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**The block decoupling problem for systems over a ring.**
*(English)*
Zbl 0964.93025

The reviewed paper is devoted to studying from a geometric point of view the decoupling problem, by means of static state feedback, for systems with coefficients in a ring. The problem has been previously considered from a transfer function point of view [cf. K. B. Datta and M. L. J. Hautus, SIAM J. Control Optimization 22, 28-39 (1984; Zbl 0529.93020)] for a system with coefficients over a unique factorization domain, also it has been studied from a geometric point of view [cf. O. Sename and J. F. Lafay, “Decoupling of linear systems with delays”, Proc. IEEE Conf. Decision and Control, Orlando, FL (1994)]. There are two main contributions in the reviewed paper. The first one consists of the introduction, for systems over a ring, of the new geometric concept of precontrollability submodule (p.c.s.). The precontrollability submodule captures the geometric aspects of the notion of the controllability submodule and is instrumental in analyzing and solving the decoupling problem. The second contribution consists of showing that the conditions given by Inaba, Ito and Munaka [cf. H. Inaba, N. Ito and T. Munaka, “Decoupling and pole assignment for linear systems defined over principal ideal domains”, Linear Circuits, Systems and Signal Processing, Sel. Pap. 8th Int. Symp. Math. Theory Network Syst., Phoenix/Ariz. 1987, 55-62 (1988; Zbl 0675.93017)] for the solvability by means of a static-state feedback of the decoupling problem for systems over a principal ideal domain can be formulated using p.c.s.’s in a less restrictive way for systems over general Noetherian rings. The important consequence of the above result is, in particular, that the solvability of the decoupling problem for the systems over rings is characterized by the same conditions as in the case of systems over a field. In addition, it is shown that, using an equivalent formulation of these conditions, the solvability of the problem can be checked without actually computing maximum precontrollability or controllability submodules.

Reviewer: Nikolai I.Osetinski (Moskva)